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Okay, let's dive deep into the dreaded "undefined symbol" error you might encounter when compiling shared libraries (often `.so` files on Linux, `.dylib` on macOS, or `.dll` on Windows) intended to be used with NumPy 2.0. This is a common and frustrating problem, but understanding the underlying reasons and having a systematic approach can help you resolve it. I'll provide a detailed tutorial with a practical example.

*Understanding the "Undefined Symbol" Error*

The "undefined symbol" error, in its simplest form, means that the linker (the program that combines compiled object files into a shared library or executable) cannot find the definition of a function or variable that your code is trying to use. Think of it like this: You have a Python script that imports a function called `my_function` from a shared library, but when the library was built, the definition of `my_function` was not properly included or linked.

Specifically, when compiling a shared library intended for use with NumPy, the most common culprits are:

1. *Missing NumPy API Declarations:* You're using NumPy C API functions (like `PyArray_SimpleNew`, `PyArray_DATA`, `PyModule_Create2`, etc.) but haven't properly included the necessary header files and linked against the NumPy library.
2. *Incorrect Linker Flags:* The linker doesn't know where to find the NumPy library. You haven't provided the right `-I` (include directory) and `-L` (library directory) flags, or you haven't specified `-l` (library name) for NumPy.
3. *ABI (Application Binary Interface) Incompatibilities:* The version of NumPy you used to compile the library might be different from the version of NumPy that's installed and being used by your Python interpreter. This is especially likely to be an issue if you're compiling on one machine and deploying on another or when upgrading NumPy.
4. *Symbol Visibility Issues (Hidden Symbols):* In some cases, symbols might be "hidden" during compilation. This means they're not ex ...

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